588 



SCIENCE. 



fN. S. Vol. IX. No. 225. 



It is not a number oo we are ever concerned 

 with. Wtien we say w (a) = oo we really mean 

 lira I iv{z) | = oo , .r = a. Again when we ask how 

 does w{z) behave for 2=00 we really mean 

 how does w (1 / f) beha\(e in the vicinity of 

 f = o where c ^ 1 / 2. Thereby c is never re- 

 quired to assume the value of o. On using 

 the sphere instead of the plane we get the 

 punktierte Kugel. The missing f)oiut we can 

 supply or not at our option. In any case 

 no number shall correspond to it. We firmly 

 believe that the easy intuitional way of tr^at. 

 ing 00 in the function theory of a complex 

 variable must be modified aa here indicated. 



The last chapter is devoted to a brief apergu 

 of the function theory from the standpoint of 

 Cauchy aud Riemaun. We cannot appreciate 

 the difficulties mentioned in g 164 as underlying 

 the definition of a function from the Cauchy- 

 Riemanu standpoint. They seem to us to be 

 due to the belief on the part of the authors 

 that we must take the whole z-jjlane into our 

 definition from this point of view. Such is not 

 the case. As a domain D for the variable s we 

 take any point multiplicity consisting only of 

 interior points. If it be possible to pass from 

 any point of D to any other of it along a con- 

 tinuous curve x = 4,{l), yz=f(t) we say i) is a 

 simple domain. Otherwise D is composed of 

 simple domains JD = Dj + A + . • • To get a 

 synectic function w{z) for D we take two single 

 valued functions u{x,y), v(x,y) defined over D 

 and such that for every point in D they have a 

 total differential and satisfy the equation. 









In any one of these simple regions as R^, 

 w (z) can be developed into an integral positive 

 power series. The analytic function / (z) ob- 

 tained from one of these elements is identical 

 with IV (2). There certainly is no reason to sup- 

 pose that / (2) when continued into another re- 

 gion i2/3 should be identical with w (2) in this 

 region. This seems to answer all the objections 

 in I and II of this article. Indeed, the advan- 

 tage seems to be decidedly on the side of Cauchy, 

 for exactly one of the points urged against 

 Cauchy's theory is now without force, while it 

 is, indeed, an important matter from Weierstrass' 



standpoint. This, in the author's words, is : 

 "That Cauchy's definition implies in various 

 ways a considerable preliminary grasp of the 

 logical possibilities attached to the study of sin- 

 gular points." From our standpoint we fix in 

 advance the domain D ; it has no more singular 

 points than we choose to assign. Not so with 

 the analytic function. Here an element is given, 

 one singular point must lie on its circle of con- 

 vergence. Where the others are is a subject of 

 further study. 



We cannot see the difficulty mentioned under 

 III. It is, ineeed, an interesting matter to know 

 ' the irreducible minimum of conditions to im- 

 pose on tv (2),' but it seems to us nowise neces- 

 sary. It suffices that we know the necessary 

 and sufiioient conditions in order that iv (z) 

 can be developed according to Taylor's Theo- 

 rem. This we know and we have taken them 

 into our definition of w (z). It may be inter- 

 esting to remark, however, that these conditions 

 are already known, as will appear in a remark- 

 able paper of E. Goursat shortly to be published. 



We close, congratulating the authors for writ- 

 ing a woi'k which we believe will prove an ex- 

 cellent aid to acquire some of the essentials of 

 the theory of function. We should have pre- 

 ferred to see the two theories of Cauchy and 

 Weierstrass blended together into an organic and 

 indivisible whole. Although these two theories 

 grew up quite distinct, thej' have already 

 been welded into one greater and more power- 

 ful theory. It is only the purist who still tena- 

 ciously clings to the methods of Weierstrass. It 

 seems, therefore, very desirable to us that an 

 introductory work should be written more in 

 accordance with this fact. 



James Pierpont. 



Yale University, March, 1899. 



A Handbook of Metallurgy. By De. Carl 

 SCHNABEL. Translated by Henry Louis. 

 New York, The Macmillan Company. Two 

 volumes, medium 8vo. Total pages, 1608. 

 Illustrated. Volume 1, copper, lead, silver, 

 gold. Volume 2, zinc, cadmium, mercury, 

 bismuth, tin, antimony, arsenic, nickel, co- 

 balt, platinum, aluminum. Price, $10.00. 

 The author states in the preface that, while 



many exhaustive works have appeared on the 



